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1 Growth Theory through the Lens of Development Economics Abhijit Banerjee and Esther Duflo Massachusetts Institute of Technology Abstract Growth theory traditionally assumed the existence of an aggregate production function, whose existence and properties are closely tied to the assumption of optimal resource allocation within each economy. We show extensive evidence, culled from the microdevelopment literature, demonstrating that the assumption of optimal resource allocation fails radically. The key fact is the enormous heterogeneity of rates of return to the same factor within a single economy, a heterogeneity that dwarfs the cross-country heterogeneity in the economy-wide average return. Prima facie, we argue, this evidence poses problems for old and new growth theories alike. We then review the literature on various causes of this misallocation. We go on to calibrate a simple model which explicitly introduces the possibility of misallocation into an otherwise standard growth model. We show that, in order to match the data, it is not enough to have misallocated factors: there also needs to be important fixed costs in production. We conclude by outlining the contour of a possible non-aggregate growth theory, and review the existing attempts to take such a model to the data. JEL numbers O0, O10, O11, O12, O14, O15, O16, O40 Keywords: Non-aggregative growth theory; aggregate production function; factor allocation; non-convexities.

2 Growth Theory through the Lens of Development Economics Abhijit V. Banerjee and Esther Duflo December Introduction: Neo-classical Growth Theory The premise of neo-classical growth theory is that it is possible to do a reasonable job of explaining the broad patterns of economic change across countries, by looking at it through the lens of an aggregate production function. The aggregate production function relates the total output of an economy (a country, for example) to the aggregate amounts of labor, human capital and physical capital in the economy, and some simple measure of the level of technology in the economy as a whole. It is formally represented as F(A, K P K H, L) where K P and K H are the total amounts of physical and human capital invested, L is the total labor endowment of the economy and A is a technology parameter. The aggregate production function is not meant to be something that physically exists. Rather, it is a convenient construct. Growth theorists, like everyone else, have in mind a world where production functions are associated with people. To see how they proceed, let us start with a model where everyone has the option of starting a firm, and when they do, they have access to an individual production function Y = F (K P, K H, L, θ), (1) where K P and K H are the amounts of physical and human capital invested in the firm and L is the amount of labor. θ is a productivity parameter which may vary over time, but at any point of time is a characteristic of the firm s owner. Assume that F is increasing in all its inputs. To make life simpler, assume that there is only one final good in this economy and physical capital is made from it. Also assume MIT, Department of Economics, 50 Memorial Drive, Cambridge, MA For financial support, the authors are grateful to the National Science Foundation under the grant SES (Banerjee), the Alfred P. Sloan Foundation (Duflo) and the John D. and Catherine MacArthur Foundation. We are also grateful to Pranab Bardhan, Michael Kremer, Rohini Pande, Chris Udry and Ivan Werning for helpful conversations, to Philippe Aghion and Seema Jayachandran for detailed comments, and to Charles Cohen and Thomas Wang for excellent research assistance. A part of this material was presented as the Kuznets Memorial Lecture, 2004, at Yale University. We are grateful for the many comments that we received from the audience. 1

3 that the population of the economy is described by a distribution function G t (W, θ), the joint distribution of W and θ, where W is the wealth of a particular individual and θ is his productivity parameter. Let G(θ) be the corresponding partial distribution on θ. The lives of people, as often is the case in economic models, is rather dreary: In each period, each person, given his wealth, his θ and the prices of the inputs, decides whether to set up a firm, and if so how to invest in physical and human capital. At the end of the period, once he gets returns from the investment and possibly other incomes, he consumes and the period ends. The consumption decision is based on maximizing the following utility function: δ t U(C t, θ), 0 < δ < 1. (2) t=0 1.1 The Aggregate Production Function The key assumption behind the construction of the aggregate production function is that all factor markets are perfect, in the sense that individuals can buy or sell as much as they want at a given price. With perfect factor markets (and no risk) the market must allocate the available supply of inputs to maximize total output. Assuming that the distribution of productivities does not vary across countries, we can therefore define F(K P, K H, L) to be: max { F (K P (θ), K H (θ), L(θ), θ)d G(θ)} {K P (θ),k H (θ),l(θ)} θ subject to K P (θ)dθ = K P, K H (θ)dθ = K H, and L(θ)dθ = L. θ θ θ This is the aggregate production function. It is notable that the distribution of wealth does not enter anywhere in this calculation. This reflects the fact that with perfect factor markets, there is no necessary link between what someone owns and what gets used in the firm that he owns. The fact that G(θ) does not enter as an argument of F(K P, K H, L) reflects our assumption that the distribution of productivities does not vary across countries. It should be clear from the construction that there is no reason to expect a close relation between the shape of the individual production function and the shape of the aggregate function. Indeed it is well known that aggregation tends to convexify the production set: In other words, the aggregate production function may be concave even if the individual production functions are not. In this environment where there are a continuum of firms, the (weak) concavity of the aggregate production function is guaranteed as long as the average product of the inputs in the individual production functions is bounded in the sense that there is a λ such that F (λk P, λk H, λl, θ) λ (K P, K H, L, θ) for all K P, K H, L and θ. It follows that the concavity of the individual functions is sufficient for the concavity of the aggregate but by no 2

4 means necessary: The aggregate production would also be concave if the individual production functions were S-shaped (convex to start out and then becoming concave). Alternately, the individual production function being bounded is enough to guarantee concavity of the aggregate production function. Moreover, the aggregate production function will typically be differentiable almost everywhere. It is a corollary of this result that the easiest way to generate an aggregate production function with increasing returns is to base the increasing returns not on the shape of the individual production function, but rather on the possibility of externalities across firms. If there are sufficiently strong positive externalities between investment in one firm and investment in another, increasing the total capital stock in all of them together will increase aggregate output by more (in proportional terms) than the same increase in a single firm would raise the firm s output, which could easily make the aggregate production function convex. This is the reason why externalities have been intimately connected, in the growth literature, with the possibility of increasing returns. The assumption of perfect factor markets is therefore at the heart of neo-classical growth theory. It buys us two key properties: The fact that the ownership of factors does not matter, i.e., that an aggregate production function exists; and that it is concave. The next sub-section shows how powerful these two assumptions can be. 1.2 The Logic of Convergence Assume for simplicity that production only requires physical capital and labor and that the aggregate production function, F(K p, L) defined as above, exhibits constant returns and is concave, increasing, almost everywhere differentiable and eventually strictly concave, in the sense that F < ε < 0, for any K p > K p. As noted above, this does not require the individual production functions to have this shape, though it does impose some constraints on what the individual functions can be like. It does however require that the distribution of firm-level productivities is the same everywhere. Under our assumption that capital markets are perfect, in the sense that people can borrow and lend as much as they want at the common going rate, r t, the marginal returns to capital must be the same for everybody in the economy. This, combined with the preferences as represented by (2), has the immediate consequence that for everybody in the economy: U (C t, θ) = δr t U (C t+1, θ). It follows that everybody s consumption in the economy must grow as long as δr t > 1 and shrink if δr t < 1. And since consumption must increase with wealth, it follows that everyone must be getting richer if and only if δr t > 1, and consequently the aggregate wealth of the economy must be growing as long as δr t > 1. In a closed economy, the total wealth must be equal to the total capital stock, and 3

5 therefore the capital stock must also be increasing under the same conditions. Credit market equilibrium, under perfect capital markets, implies that F (K P t, L) = r t. The fact that F is eventually strictly concave implies that as the aggregate capital stock grows, its marginal product must eventually start falling, at a rate bounded away from 0. This process can only stop when δf (K P t, L) = 1. As long as the production function is the same everywhere, all countries must end up equally wealthy The logic of convergence starts with the fact that in poor countries, capital is scarce, which combined with the concavity of the aggregate production function implies that the return on the capital stock should be high. Even with the same fraction of these higher returns being reinvested, the growth rate in the poorer countries would be higher. Moreover, the high returns should encourage a higher reinvestment rate, unless the income effect on consumption is strong enough to dominate. Together, they should make the poorer countries grow faster and catch up with the rich ones. Yet poorer countries do not grow faster. According to Mankiw, Romer and Weil (1992), the correlation between the growth rate and the initial level of Gross Domestic Product is small, and if anything, positive (the coefficient of the log of the GDP in 1960 on growth rate between 1960 and 1992 is ). Somewhere along the way, the logic seems to have broken down. Understanding the failure of convergence has been one of the key endeavors of the economics of growth. What we try to do in this chapter is to argue that the failure of this approach is intimately tied to the failure of the assumptions that underlie the construction of the aggregate production function and to suggest an alternative approach to growth theory that abandons the aggregate production. We start by discussing, in section 2, the two implications of the neo-classical model that are at the root of the convergence result: Both rates of returns and investment rates should be higher in poor countries. We show that, in fact, neither rates of returns nor investment are, on average, much higher in poor countries. Moreover, contrary to what the aggregate production approach implies, there are large variations in rate of returns within countries, and large variation in the extent to which profitable investment opportunities are taken advantage of. In section 3, we ask whether the puzzle (of no convergence) can be solved, while maintaining the aggregate production function, by theories that focus on reasons for technological backwardness in poor countries. We argue that this class of explanations is not consistent with the empirical evidence which suggests that many firms in poor countries do use the latest technologies, while others in the same country use obsolete modes of production. In other words, what we need to explain is less the overall technological backwardness and more why some firms do not adopt profitable technologies that are available to them (though perhaps not affordable). In section 4, we attempt to suggest some answers to the question of why firms and people in devel- 4

6 oping countries do not always avail themselves of the best opportunities afforded to them. We review various possible sources of the inefficient use of resources: government failures, credit constraints, insurance failure, externalities, family dynamics, and behavioral issues. We argue that each of these market imperfections can explain why investment may not always take place where the rates of returns are the highest, and therefore why resources may be misallocated within countries. This misallocation, in turn, drives down returns and this may lower the overall investment rate. In section 5, we calibrate plausible magnitudes for the aggregate static impact of misallocation of capital within countries We show that, combined with individual production functions characterized by fixed costs, the misallocation of capital implied by the variation of the returns to capital observed within countries can explain the main aggregate puzzles: the low aggregate productivity of capital, and the low Total Factor Productivity in developing countries, relative to rich countries. Non-aggregative growth models thus seem to have the potential to explain why poor countries remain poor. The last section provides an introduction to an alternative growth theory that does not require the existence of an aggregate production function, and therefore can accommodate the misallocation of resources. We then review the attempts to empirically test these models. We argue that the failure to take seriously the implications of non-aggregative models have led to results that are very hard to interpret. To end, we discuss an alternative empirical approach illustrated by some recent calibration exercises based on growth models that take the misallocation of resources seriously. 2 Rates of Return and Investment Rates in Poor Countries In this section, we examine whether the two main implications of the neo-classical model are verified in the data: Are returns and investment rates higher in poor countries? 2.1 Are returns higher in poor countries? Physical Capital Indirect Estimates One way to look at this question is to look at the interest rates people are willing to pay. Unless people have absolutely no assets that they can currently sell, the marginal product of whatever they are doing with the marginal unit of capital should be no less than the interest rate: If this were not true, they could simply divert the last unit of capital toward whatever they are borrowing the money for and be better off. There is a long line of papers that describe the workings of credit markets in poor countries (Banerjee 5

7 (2003) summarizes this evidence). The evidence suggests that a substantial fraction of borrowing takes place at very high interest rates. A first source of evidence is the Summary Report on Informal Credit Markets in India (Dasgupta (1989)), which reports results from a number of case studies that were commissioned by the Asian Development Bank and carried out under the aegis of the National Institute of Public Finance and Policy. For the rural sector, the data is based on surveys of six villages in Kerala and Tamil Nadu, carried out by the Centre for Development Studies. The average annual interest rate charged by professional moneylenders (who provide 45.6% of the credit) in these surveys is about 52%. For the urban sector, the data is based on various case surveys of specific classes of informal lenders, many of whom lend mostly to trade or industry. For finance corporations, they report that the minimum lending rate on loans of less than one year is 48%. For hire-purchase companies in Delhi, the lending rate was between 28% and 41%. For auto financiers in Namakkal, the lending rate was 40%. For handloom financiers in Bangalore and Karur, the lending rate varied between 44% and 68%. Several other studies reach similar conclusions. A study by Timberg and Aiyar (1984) reports data on indigenous-style bankers in India, based on surveys they carried out: The rates for Shikarpuri financiers varied between 21% and 37% on loans to members of local Shikarpuri associations and between 21% and 120% on loans to non-members (25% of the loans were to non-members). Aleem (1990) reports data from a study of professional moneylenders that he carried out in a semi-urban setting in Pakistan in The average interest rate charged by these lenders is 78.5%. Ghate (1992) reports on a number of case studies from all over Asia: The case study from Thailand found that interest rates were 5-7% per month in the north and northeast (5% per month is 80% per year and 7% per month is 125%). Murshid (1992) studies Dhaner Upore (cash for kind) loans in Bangladesh (you get some amount in rice now and repay some amount in rice later) and reports that the interest rate is 40% for a 3-5 month loan period. The Fafchamps (2000) study of informal trade credit in Kenya and Zimbabwe reports an average monthly interest rate of 2.5% (corresponding to an annualized rate of 34%) but also notes that this is the rate for the dominant trading group (Indians in Kenya, whites in Zimbabwe), while the blacks pay 5% per month in both places. The fact that interest rates are so high could reflect the high risk of default. However, this does not appear to be the case, since several of studies mentioned above give the default rates that go with these high interest rates. The study by Dasgupta (1989) attempts to decompose the observed interest rates into their various components, 1 and finds that the default costs explain 7 per cent (not 7 percentage points!) of the total interest costs for auto financiers in Namakkal and handloom financiers in Bangalore and Karur, 4% for finance companies and 3% for hire-purchase companies. The same study reports that 1 In the tradition of Bottomley (1963). 6

8 in four case studies of moneylenders in rural India they found default rates explained about 23% of the observed interest rate. Timberg and Aiyar (1984), whose study is also mentioned above, report that average default losses for the informal lenders they studied ranges between 0.5% and 1.5% of working funds. The study by Aleem Aleem (1990) gives default rates for each individual lender. The median default rate is between 1.5 and 2%, and the maximum is 10%. 2 Finally, it does not seem to be the case that these high rates are only paid by those who have absolutely no assets left. The Summary Report on Informal Credit Markets in India (Dasgupta (1989)) reports that several of the categories of lenders that have already been mentioned, such as handloom financiers and finance corporations, focus almost exclusively on financing trade and industry while Timberg and Aiyar (1984) report that for Shikarpuri bankers at least 75% of the money goes to finance trade and, to lesser extent, industry. In other words, they only lend to established firms. It is hard to imagine, though not impossible, that all the firms have literally no assets that they can sell. Ghate (1992) also concludes that the bulk of informal credit goes to finance trade and production, and Murshid (1992), also mentioned above, argues that most loans in his sample are production loans despite the fact that the interest rate is 40% for a 3-5 month loan period. Udry (2003) obtains similar indirect estimates by restricting himself to a sector where loans are used for productive purpose, the market for spare taxi parts in Accra, Ghana. He collected 40 pairs of observations on price and expected life for a particular used car part sold by a particular dealer (e.g., alternator, steering rack, drive shaft). Solving for the discount rate which makes the expected discounted cost of two similar parts equal gives a lower bound to the returns to capital. He obtains an estimate of 77% for the median discount rate. Together, these studies thus suggest that people are willing to pay high interest rates for loans used for productive purpose, which suggests that the rates of return to capital are indeed high in developing countries, at least for some people. Direct Estimates Some studies have tried to come up with more direct estimates of the rates of returns to capital. The standard way to estimate returns to capital is to posit a production function (translog and Cobb- Douglas, generally) and to estimate its parameters using OLS regression, or instrumenting capital with 2 Here we make no attempt to answer the question of why the interest rates are so high. Banerjee (2003) argues that it is not implausible that the enormous gap between borrowing and lending rates implied by these numbers, simply reflects the cost of lending (monitoring and contracting costs of various kinds). Hoff and Stiglitz (1998) suggest an important role for monopolistic competition, in the presence of a fixed cost of lending. There is also a view that the market for credit is monoploized by a small number of lenders who earn excess profits, but Aleem (1990) finds no evidence of excess profits. 7

9 its price. Using this methodology, Bigsten, Isaksson, Soderbom and Al (2000) estimate returns to physical and human capital in five African countries. They estimate rates of returns ranging from 10% to 32%. McKenzie and Woodruff (2003) estimate parametric and non-parametric relationships between firm earnings and firm capital. Their estimates suggest huge returns to capital for these small firms: For firms with less than $200 invested, the rate of returns reaches 15% per month, well above the informal interest rates available in pawn shops or through micro-credit programs (on the order of 3% per month). Estimated rates of return decline with investment, but remain high (7% to 10% for firms with investment between $200 and $500, 5% for firms with investment between $500 and $1,000). Such studies present serious methodological issues, however. First, the investment levels are likely to be correlated with omitted variables. For example, in a world without credit constraints, investment will be positively correlated with the expected returns to investment, generating a positive ability bias (Olley and Pakes (1996)). McKenzie and Woodruff attempt to control for managerial ability by including the firm owner s wage in previous employment, but this may go only part of the way if individuals choose to enter self-employment precisely because their expected productivity in self-employment is much larger than their productivity in an employed job. Conversely, there could be a negative ability bias, if capital is allocated to firms in order to avoid their failure. Banerjee and Duflo (2004) take advantage of a change in the definition of the so-called priority sector in India to circumvent these difficulties. All banks in India are required to lend at least 40% of their net credit to the priority sector, which includes small-scale industry, at an interest rate that is required to be no more than 4% above their prime lending rate. In January, 1998, the limit on total investment in plants and machinery for a firm to be eligible for inclusion in the small-scale industry category was raised from Rs. 6.5 million to Rs. 30 million. In 2000, the limit was lowered back to Rs 10 million Banerjee and Duflo (2004) first show that, after the reforms, newly eligible firms (those with investment between 6.5 million and 30 million) received on average larger increments in their working capital limit than smaller firms. They then show that the sales and profits increased faster for these firms during the same period. The opposite happened when the priority sector was contracted again. Putting these two facts together, they use the variation in the eligibility rule over time to construct instrumental variable estimates of the impact of working capital on sales and profits. After computing a non-subsidized cost of capital, they estimate that the returns to capital in these firms must be at least 74%. There is also direct evidence of very high rates of returns on productive investment in agriculture. Goldstein and Udry (1999) estimate the rates of returns to the production of pineapple in Ghana. The rate of returns associated with switching from the traditional maize and Cassava intercrops to pineapple is estimated to be in excess of 1,200%! Few people grow pineapple, however, and this figure may hide some heterogeneity between those who have switched to pineapple and those who have not. 8

10 Evidence from experimental farms also suggests that, in Africa, the rate of returns to using chemical fertilizer (for maize) would also be high. However, this evidence may not be realistic, if the ideal conditions of an experimental farm cannot be reproduced on actual farms. Foster and Rosenzweig (1995) show, for example, that the returns to switching to high yielding varieties were actually low in the early years of the green revolution in India, and even negative for farmers without an education. This is despite the fact that these varieties had precisely been selected for having high yields, in proper conditions. But they required complementary inputs in the correct quantities and timing. If farmers were not able or did not know how to supply those, the rates of returns were actually low. To estimate the rates of returns to using fertilizer in actual farms in Kenya, Duflo, Kremer and Robinson (2003), in collaboration with a small NGO, set up small scale randomized trials on people s farms: Each farmer in the trials delimited two small plots. On one randomly selected plot, a field officer from the NGO helped the farmer apply fertilizer. Other than that, the farmers continued to farm as usual. They find that the rates of returns from using a small amount of fertilizer varied from 169% to 500% depending on the year, although of returns decline fast with the quantity used on a plot of a given size. This is not inconsistent with the results in Foster and Rosenzweig (1995), since by the time this study was conducted in Kenya, chemical fertilizer was a well established and well understood technology, which did not need many complementary inputs. The direct estimates thus tend to confirm the indirect estimates: While there are some settings where investment is not productive, there seems to be investment opportunities which yield substantial rates of returns. How high is the marginal product on average? The fact that the marginal product in some firms is 50% or 100% or even more does not imply that the average of the marginal products across all firms is nearly as high. Of course, if capital always went to its best use, the notion of the average of the marginal products does not make sense. The presumption here is that there may be an equilibrium where the marginal products are not equalized across firms. One way to get at the average of the marginal products is to look at the Incremental Capital Output Ratio (ICOR) for the country as a whole. The ICOR measures the increase in output predicted by a one unit increase in capital stock. It is calculated by extrapolating from the past experience of the country and assumes that the next unit of capital will be used exactly as efficiently (or inefficiently) as the last one. The inverse of the ICOR therefore gives an upper bound for the average marginal product for the economy it is an upper bound because the calculation of the ICOR does not control for the effect of the increases in the other factors of production which also contributes to the increase in output. 3 For the 3 The implicit assumption that the other factors of production are growing is probably reasonable for most developing 9

11 late 1990s, the IMF estimates that the ICOR is over 4.5 for India and 3.7 for Uganda. The implied upper bound on the average marginal product is 22% for India and 27% in Uganda. This is also consistent with the work of Pessoa, Cavalcanti-Ferreira and Velloso (2004) who estimate a production function using cross-country data and calculate marginal products for developing countries which are in the 10-20% range. It seems that the average returns are actually not much higher than the 9% or so, which is the usual estimate for the average stock market return in the US. Variations in the marginal products across firms. To reconcile the high direct and indirect estimates of the marginal returns we just discussed and an average marginal product of 22% in India, it would have to be that there is substantial variation in the marginal product of capital within the country. Given that the inefficiency of the Indian public sector is legendary, this may just be explained by the investment in the public sector. However, since the ICOR is from the late 1990s, when there was little new investment (or even disinvestment) in the public sector, there must also be many firms in the private sector with marginal returns substantially below 22%. The micro evidence reported in Banerjee (2004), which shows that there is very substantial variation in the interest rate within the same sub-economy, certainly goes in this direction. The Timberg and Aiyar (1984) study mentioned above, is one source of this evidence: It reports that the Shikarpuri lenders charged rates that were as low as 21% and as high as 120%, and some established traders on the Calcutta and Bombay commodity markets could raise funds for as little as 9%. The study by Aleem (1990), also mentioned above, reports that the standard deviation of the interest rate was 38.14%. Given that the average lending rate was 78.5%, this tells us that an interest rate of 2% and an interest rate of 150% were both within two standard deviations of the mean. Unfortunately, we cannot quite assume from this that there are some borrowers whose marginal product is 9% or less: The interest rate may not be the marginal product if the borrowers who have access to these rates are credit constrained. Nevertheless, given that these are typically very established traders, this is less likely than it would be otherwise. Ideally we would settle this issue on the basis of direct evidence on the misallocation of capital, by providing direct evidence on variations in rates of return across groups of firms. Unfortunately such evidence is not easy to come by, since it is difficult to consistently measure the marginal product of capital. However, there is some rather suggestive evidence from the knitted garment industry in the Southern Indian town of Tirupur (Banerjee and Munshi (2004); Banerjee, Duflo and Munshi (2003)). Two groups of people operate in Tirupur: the Gounders, who issue from a small, wealthy, agricultural community from the area around Tirupur, who have moved into the ready-made garment industry because there was not much investment opportunity in agriculture. Outsiders from various regions and communities countries, except perhaps in Africa. 10

The (mis)allocation of capital Abhijit V. Banerjee Esther Duflo Kaivan Munshi September, 2002 Abstract Is capital allocated so that its marginal product is equated to the market interest rate? Is the marginal

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